Novel Numerical Investigations of Fuzzy Cauchy Reaction–Diffusion Models via Generalized Fuzzy Fractional Derivative Operators

Abstract

The present research correlates with a fuzzy hybrid approach merged with a homotopy perturbation transform method known as the fuzzy Shehu homotopy perturbation transform method (SHPTM). With the aid of Caputo and Atangana–Baleanu under generalized Hukuhara differentiability, we illustrate the reliability of this scheme by obtaining fuzzy fractional Cauchy reaction–diffusion equations (CRDEs) with fuzzy initial conditions (ICs). Fractional CRDEs play a vital role in diffusion and instabilities may develop spatial phenomena such as pattern formation. By considering the fuzzy set theory, the proposed method enables the solution of the fuzzy linear CRDEs to be evaluated as a series of expressions in which the components can be efficiently identified and generating a pair of approximate solutions with the uncertainty parameter λ∈[0,1]. To demonstrate the usefulness and capabilities of the suggested methodology, several numerical examples are examined to validate convergence outcomes for the supplied problem. The simulation results reveal that the fuzzy SHPTM is a viable strategy for precisely and accurately analyzing the behavior of a proposed model.

Solving Linear and Nonlinear Fuzzy Fractional Volterra-Fredholm Integro Differential Equations Using Shehu Adomian Decomposition Method

“…Nagwa et al (8) proposed the usage of the fuzzy Adomian decomposition method for solving some fuzzy fractional partial differential equations. Alqudah et al (9) introduced the novel numerical investigations of fuzzy Cauchy Reaction-Diffusion models via generalized fuzzy fractional derivative operators. In (10) the modified Adomian decomposition method (MADM) is used to determine the solution of the fuzzy fractional order Volterra-Fredholm integro-differential equations.…”

Revisiting (2+1)-dimensional Burgers’ dynamical equations: analytical approach and Reynolds number examination

“…On the other hand, a number of computational approaches are also visible for solving different evolution equations. These approaches are not limited to the Adomian decomposition method [38,39], the collective variable approach [40], the homotopy analysis method [41,42], and the homotopy-Laplace perturbation approach [43] to review a few.…”

“…However, motivated by the recent work of Hussain et al [6] on the survey and analytical-numerical implementations for various forms of multi-dimensional nonlinear Burgers equations, the present study makes use of an efficient analytical approach by the name generalized Riccati equation approach [7] to securitize the class of (2+1)-dimensional Burgers' equations by revealing yet additional sets of analytical structures to the governing scalar [44] and vector-coupled [45] Burgers' equations. Besides, the importance of the study of Burgers' equation cannot be overemphasized, as various mathematical physics models could suitably be deduced from Burgers' equation expressed in equation (1), including the unsteady-state diffusion equation, steady-state diffusion equation, inviscid Burgers' equation, damped diffusion equation -when the w x ¶ ¶ is replaced by a real constant-, and the advection/convection-diffusion equation [34,35,42] among others; in short, the study of Burgers' equations paves a way for the study and analysis of several mathematical models of interesting relevance. Furthermore, the approach of concern (the generalized Riccati equation approach [7]) has been proven to divulge various sets of soliton solutions when applied on Schr ö dinger equations, and different exact solutions, including hyperbolic function solutions, periodic function solutions, irrational function solutions and other forms of exact solutions when applied on real-valued evolution equations; read relatively recent works by AlQarni et al [8] and Aljohani et al [9].…”

Exact Solutions of the Nonlinear Modified Benjamin-Bona-Mahony Equation by an Analytical Method

“…For more on analytical methods, see [22][23][24][25][26][27][28]. Furthermore, we mention some of the well-known computational methods for approximate soliton solutions here, including the famous Adomian decomposition approach [29], the Laplace-homotopy perturbation method [30] and the homotopy analysis approach [31], among others.…”